Analysis and Probability seminar

Abstract: Inductive limits are a central construction in operator algebras because they allow one to construct sophisticated objects using well-understood building blocks, particularly matrix algebras. However, if one wants to use matrix algebras as building blocks, then this construction can be quite restrictive, at least in the case of C*-algebras (closed *-algebras of bounded Hilbert space operators). In particular, the only C*-algebras that arise as inductive limits of matrix algebras are the almost finite dimensional (AF) algebras. This well-behaved class includes interesting examples such as the CAR algebra but falls far short of covering all amenable C*-algebras (a notably tractable class including noncommutative tori, and Cuntz and Toeplitz algebras). To realize this broader class as inductive limits of matrix algebras, one must consider inductive sequences in a broader category: the category of operator systems (*-subspaces of C*-algebras). In this talk, we will describe inductive sequences in the classical sense and then delve into the broader context of operator systems where we will investigate amenability and when one can still expect a C*-algebra at the limit.

Part of this is based on joint work with Niklas Galke, Lauritz van Luijk, Matt Kennedy, and Alexander Stottmeister.